/mg/ maths general: 3 edition

[Anonymous No.10987515]

talk maths, formerly >>10973904

http://bristol.ac.uk/maths/news/2019/number-3.html
[math]569936821221962380720^3 + (-569936821113563493509)^3 + (-472715493453327032)^3 = 3[/math]

[Anonymous No.10987808]

>>10987515
What's the point of finding those?

[Anonymous No.10987824]

>>10987808
A lot of math is done for the sake of it. But eventually some piece of math, some little doodle in a journal somewhere is going to be useful to someone.

[Anonymous No.10987875]

Any good postgrad cryptography/astronomy in Europe?

[Anonymous No.10987884]

>>10987875
Oxbridge, Imperial.

[Anonymous No.10987890]

>>10987824
How do I make these things useful?

[Anonymous No.10987896]

Terry Tao has proved that almost all starting points in the Collatz process comes very close to one!

https://arxiv.org/abs/1909.03562

[Anonymous No.10987900]

>>10987896
Oh wait I'm late to the party. Seems like it was posted already in the other /mg/ thread.

[Anonymous No.10988331]

Should I go with Artin or Dummit/Foote for algebra? Which one is better for self-learning?
Also is it easier to start with algebra or basic analysis for someone with little mathematical maturity?

[Anonymous No.10988630]

So what was your start in your university career like? What was your level of knowledge?
I enrolled in math, but I worry that ill fail hard because
1. In highschool (Europe) i wasnt among the best math students. Generally i didnt put effort into any classes, but the Grades where still very good. Except for math, the grades where only average. So i worry i dont have that special "it factor"/or lack the talent to Excel in university
2. I never did anything close to university math. So i worry that i have no clue whats going on and only play catch up...

[Anonymous No.10988652]

>>10988630
>So i worry i dont have that special "it factor"/or lack the talent to Excel in university
literally a meme
I had an 8.3 average in math also but just I was retarded and weren't rigorous enough with my studying

[Anonymous No.10988656]

Starting at Monday, I will have a 1.5, 2 hour commute to uni every day, if I decide to go. What is something productive to do during that time?

[Anonymous No.10988660]

>>10988656
COOOOOM

[Anonymous No.10988674]

>>10987515
What's the explanation that their report two "major finds" so close to each other?

[Anonymous No.10988675]

>>10987890
Diophantine indeterminacy is used extensively in the field of cryptography. Corollarilally, mapping in Diophantine space is probably useful for breaking the concept of cryptography itself.

[Anonymous No.10988798]

[math]g(x) = \sqrt(9 - |x-1|) [/math]

how do I solve this to show range algebraically? I know it's [math]0\leq y \leq 3 [/math]

[Anonymous No.10988808]

>>10988798
if g is to be a real function, then the expression inside the square root must be non negative. So put 9-|x-1| >= 0 and solve for x

[Anonymous No.10988818]

>>10988808
I already did that in step 1 to show domain.

[math]-8\leq x \leq 10[/math]

[Anonymous No.10988851]

>>10988818
Ah, range. So put g(x) = y and solve for x
y^2 = (9-|x-1|) (note that we are taking the positive square root, so y^3 is also positive) break it into the relevant cases and analyse the resulting curve.
For -1 < x < 0 we have y^2 = 9 - x +1 = 10 - x
Or y = sqrt(10-x)
That gives sqrt(10) y < sqrt(11) (the sqrt(K-x) is a decreasing function function)
Now do x>= 0

[Anonymous No.10989023]

>>10988630
I was a really average student in math in high school because I didn't put any effort in it. I think about going to university next year so I'll take advantage of current year to catch up.

>>10988652
Are you studying uni math now ? How is it going ?


Also can someone explain this chart ? Does the book on the left cover all the material listed on the right ? Also what is the name of the first book ?

[Anonymous No.10989094]

>>10989023
>Does the book on the left cover all the material listed on the right ?
No, the books cover some of the topics only

>Also what is the name of the first book?
https://www.amazon.com/Linear-Algebra-Geometry-Logic-Applications/dp/2881246834

[Anonymous No.10989104]

>>10989094
Thank you pal

[Anonymous No.10989204]

>>10988630
I am Not going to pretend that your Highschool grades mean shit, but why study math in the first place?
If you consider it "the thing you want to do" then nothing could be less meaningful then your inability to motivate yourself to do the stupid and boring calculations which are Highschool math and nothing could be more meaningful then your genuine interest in the subject.

> I never did anything close to university math
That is also true for almost everybody around you, much more important is motivation and willingness to put in the work.

[Anonymous No.10989231]

>>10988674
>What's the explanation that their report two "major finds" so close to each other?
Availability of computational power.

[Anonymous No.10989237]

>>10988656
Read a book.

[Anonymous No.10989496]

>>10989023

check the wikia, that's a meme chart

[Anonymous No.10989502]

>>10989237
on commute-ative algebra

[Anonymous No.10989625]

>>10987824
>A lot of math is done
They used fucking computers

[Anonymous No.10989653]

>>10989496
>that's a meme chart
What do you mean?

[Anonymous No.10989928]

>>10989653
It's a joke to trick undergrads into getting really hard books so that they feel stupid when they don't understand anything.


Get Enderton's elements, or his introduction to logic if you like it more, after having a nice foundation most intro books will be at your reach.

[Anonymous No.10989950]

>>10989502
Underrated post

[Anonymous No.10990073]

Suppose I've got some natural numbers a,b,c,d such that a*b=c*d and gcd(a,b)=gcd(c,d)=1. Does this tell us anything about a,b,c and d?

[Anonymous No.10990267]

>>10990073
not really no
2*3 = 6*1

[Anonymous No.10990275]

>>10989625
And?
The hard part were obviously the algorithms, it's not like the trivial approach had any chance to succeed.

[Anonymous No.10990294]

>>10987896
>hurr almost all
>durr they come close to 1 but maybe not 1
fuck this shit

[Anonymous No.10990319]

>>10990294
>Hurr why would you proof anything at all which isn't the big result

[Anonymous No.10990349]

Im looking for concepts and techniques that i can apply to help me unterstand Basic up to advanced math concepts.

[Anonymous No.10990392]

>>10990073
not much
a small useful fact is that you can rewrite the numbers as
a = xy
b = zt
c = xz
d = yt
where x, y, z, t are pairwise coprime

[Anonymous No.10990473]

Can someone intuitively explain to me what this statement means?
>Today there's a 31% likelihood of precipitation.

Does it mean during the span of 24 hours, it will rain for 31% of that? I don't know, but probability theory sounds like a hard cope to me.

[Anonymous No.10990481]

>>10990473
We measure how good the forecast is by its "reliability" - a perfectly reliable probability forecast of 31 percent means that on the average, it rains somewhere within the space-time value 31 percent of the time when that 31 percent probability forecast is made. When it rains, of course, we should expect higher probability values, and lower values when it doesn't rain. Perfect forecasting would consist of 100 percent probabilities only when it rains, and zero probabilities when it fails to rain, but that level of certainty is impossible (for many reasons). Thus, it rains one time out of ten when the probability forecast is for 10 percent (assuming reliable forecasting).

[Anonymous No.10990525]

Is there a purely algebraic proof that [math]\mathbb{C}[/math] is algebraically closed? By purely algebraic I mean that it only uses results from algebra, i.e field, group and galois theory.

[Anonymous No.10990534]

>>10989496

I'm >>10989023 I know it's a meme chart, but since I see it so often on /sci/ I was just curious to understand how it works.

[Anonymous No.10990558]

>>10990525
Afaik, no.
You need at least Bolzano's theorrm

[Anonymous No.10990591]

Let f: A --> [math] \R [/math] and g: B --> [math] \R [/math] be functions

What's the domain of f-g and f/g? Math has never been my strong subject, and these non-number questions are the absolute death of me since I have basically 0 intuition about math.

How can I know what the domain is without having an actual equation, since the domain is depended on the values of f and g (?)

[Anonymous No.10990597]

>>10990591
f-g is A intersection B and f/g is a A intersection B minus all points where g becomes 0

[Anonymous No.10990610]

>>10990597
f - g = [math] A \cap B [/math]

f/g = [math] A \cap B, g \neq 0 [/math]

something like this?

[Anonymous No.10990615]

>>10990610
Yeah. You need both f and g to be defined for either f-g and f/g to exist. But if the quotient is to exist, then also g\neq 0

[Anonymous No.10990660]

Could someone check my work?
I'm to prove that a splitting field [math]K[/math] of an irreducible separable polynomial [math]f(x)\in F\left [ x \right ] [/math] of degree [math]n[/math] has a Galois group with [math]\left | Gal(K/F) \right |[/math] divisible by [math]n[/math]. First, since [math]K[/math] is finite normal and separable, it's Galois over [math]F[/math], and by the fundamental theorem of Galois theory, it follows that [math]\left | Gal(K/F) \right |=\left [ K:F \right ][/math]. Now [math]K=F\left ( \alpha _{1},...,\alpha _{n} \right )[/math], so [math]\left [ K:F \right ]=\left [ K:F\left ( \alpha _{1},...,\alpha _{n-1} \right ) \right ]\cdot \cdot \cdot [F(\alpha_{1}):F]= \left [ K:F\left ( \alpha _{1},...,\alpha _{n-1} \right ) \right ]\cdot \cdot \cdot n[/math] since [math][F(\alpha_{1}):F]=deg(\alpha_{1},F)=deg(f)=n[/math]. Then it follows that [math]n[/math] divides [math]\left | Gal(K/F) \right | [/math].

[Anonymous No.10990689]

>>10990349
Have your tried learning math?

[Anonymous No.10990706]

>>10990525
No, because C isn't a purely algebraic object and can't be defined in a purely algebraic way.

[Anonymous No.10990711]

>>10990660
looks good to me

[Anonymous No.10990715]

>>10990392
>>10990267
Damn, well thanks boys. Forgot to mention that gcd(a,c) was also 1, but considering>>10990267
I doubt that will really help. The variables really break up into bigger pieces that should only have one valid case, so I broke it up into different possibilities, but one didn't resolve where I assumed a|d, which also ended up being the same thing as saying c|b.

[Anonymous No.10990776]

>>10990706
How so?
By Cardano'a formula every real polynomial degree 3 has at least one real root. Then apply the argument in (replacing the reference to the intermediate value theorem to the existence of roots established by Cardano's formula) https://math.stackexchange.com/a/1587997 , this should yield a full proof, no?
Maybe I'm retarded, but I don't see why the FTA would depend on the continuity of the polynomial or completeness of the reals.
Like, the algebraic closure of Q is algebraically closed without being complete.

[Anonymous No.10990794]

>>10990776
FTA doesn't depend on continuity but C is an analytic object so you can't even describe it without analysis. You may be mistaking C for the algebraic closure of Q, which is a smaller set than C but has every root of every polynomial.

C just has the property that without defining it as the literal closure, you can prove FTA in it. But you could just as easily just break Q by every polynomial and ignore analysis entirely.

[Anonymous No.10990807]

>>10990776
The algebraic closure of Q is a purely algebraic object.
>cardano's formula
>implying you can show that the complex numbers have all square roots without appealing to completeness
The thing is, you have an analytic object, you can't prove algebraic facts about it without exploiting the analytic data you have on it.

[Anonymous No.10990844]

>>10987515
can anyone help me with understanding eigenvectors/values

if [math]Av = \lambdav [/math]
and I find [math]\lambda_1 v_1 ,\lambda_2v_2[\math].

Shouldnt
[math]\lambda_1 v_1 = \lambda_2v_2[\math]
???????

[Anonymous No.10990849]

>>10990844
Eigenvectors corresponding to different eigenvalues are necessarily linearly independent, so you cannot have, as you claim, [math]\lambda_{1}v_{1}=\lambda_{2}v_{2}[/math] unless [math]\lambda_{1}=\lambda_{2}[/math].

[Anonymous No.10990858]

Does the order of a transitive subgroup of [math]S_{n}[/math] necessarily divide [math]n[/math]?

[Anonymous No.10990860]

>>10990858
oops, meant to ask is the order always divisible by [math]n[/math]

[Anonymous No.10990873]

>>10990844
>Shouldnt
>[math]\lambda_1 v_1 = \lambda_2v_2[\math]
Why would that be the case?

[Anonymous No.10990877]

>>10990525
The elements of C are defined by supremum / limiting processes.
Attempt to describe C algebraically.

[Anonymous No.10990880]

>>10990858
If by transitive you mean a subgroup that acts transitively on {1,...,n}, then yes because of the orbit-stabilizer formula

[Anonymous No.10990881]

>>10990776
R is an analytic object you dolt.
Again, any statement you make about R is an analytic statement. Whether or not you think you're using completeness, you are. You are always, always using the least upper bound property. When you add numbers in R you are using the least upper bound property.

[Anonymous No.10990889]

>>10990877
>The elements of C are defined by supremum / limiting processes.
Well, you can define lR purely axiomatically though, so you do not need limits.

But as pointed out by >>10990881 you still inherently need analytical concepts to even define it.

[Anonymous No.10990893]

>>10990889
He said limiting OR supremum.
Axioms use supremum.

[Anonymous No.10990899]

Goes from primary school maths straight into
BTFO-ing infinitists. Yeah, BASED wildburger.

[Anonymous No.10990900]

>>10990893
Oh I am blind, yes of course.

[Anonymous No.10990903]

>>10988331
dummit/foote, friend.

but if you actually don't have much mathematical maturity, i recommend starting with basic, naive set theory

[Anonymous No.10990905]

>>10990899
The nature of philosophical questions is that you can have very easily a strong opinion on them without having any real knowledge of the subject they are concerned with.

[Anonymous No.10990906]

>>10990858
>>10990860
Let [math]G[/math] be a transitive subgroup of [math]S_n[/math] acts on [math]\{1 \dots n\}[/math], take any [math]x \in \{1 \dots n\}[/math].
Define [math]G_i := \{ g \in G : g(x) = i\}[/math] for i = 1...n.
For any i, j there is element h of G such that h(i) = j. Then the map [math]g \mapsto hg[/math] is an injection from [math]G_i[/math] to [math]G_j[/math].
Hence all [math]G_i[/math] have same size, so [math]|G| = n |G_1|[/math].

[Anonymous No.10990919]

>>10990889
And one of the axioms is the least upper bound property, which is the purest, most distilled analytic property one can deign to find.
The real numbers are inextricably analytic.

[Anonymous No.10990922]

>>10990919
>>10990889
Oh, I didn't read the rest of your post where you agreed with me. Oops!

[Anonymous No.10990927]

>>10990922
>>10990919
And apparently I can't read.>>10990893

[Anonymous No.10990930]

>>10990906
This helped, thanks anon.

[Anonymous No.10990947]

Physics fag here, why is complex analysis "nicer" or less than real analysis?
What does that mean about the real set and complex set?

For me it was intuitive, it was taught by Dana Fine at UMD who is basically the final boss of math there. It felt like a natural extension of vector calc after working with vectors from qm and e&m.

[Anonymous No.10990961]

>>10990947
Because complex differentiability trivially implies Cauchy-Riemann, and Cauchy-Riemann implies a form of "harmonicity", and harmonic functions are naisu.

[Anonymous No.10990974]

>>10990947
Real analysis tries to generalize away from "nice" concepts like "continuity" or "Riemann integrability" which is a central motivation, to gather the greatest space of things you can integrate (and even tries to generalize differentiability on that space), while the notions of complex analysis lead to "nicer" concepts such as holomorphic functions which have even more properties then "normal" calculus functions.

[Anonymous No.10990986]

>>10990947
Holomorphic functions are harmonic in the "right direction," i.e. they're functions with the derivative [math] \frac{\partial}{\partial \overline{z}} [/math] identically 0. Think about it, if a given function has some derivative that's identically 0, you usually think of it as constant in that direction. So this is an extremely strong property.
So complex analysis is not just nice because of C, it's nice because the subset of functions which you study is very restrictive and so they all can have very nice properties.

[Anonymous No.10990996]

I just finished high-school and want to study maths further in my own time, kind of as a hobby
What should I proceed to? I've nailed down pre-calc and everything you do in highschool, what would be the next step?

[Anonymous No.10991004]

>>10990986
nice, thanks

[Anonymous No.10991024]

>>10990996
Maybe pick up Velleman's "How to Prove It: A Structured Approach". This should cover all the basics, and you'll be pretty much ready to grab any first-year undergrad textbook and go off on your own. I'd suggest not studying too many areas at once, limit yourself to maybe 2-3. You can find all the books you need on libgen, so save your money and download them from there.
Some concrete recommendations, I'd suggest Abbot's "Understanding Analysis" if you want a easy to ready introductory book on analysis, for algebra I'd go for Fraleigh's "A First Course in Abstract Algebra". Both of these are more than accessible after you've gone through Velleman's book.

[Anonymous No.10991158]

>>10990996
>What should I proceed to? I've nailed down pre-calc and everything you do in highschool, what would be the next step?

[Anonymous No.10991168]

>>10991024
>Maybe pick up Velleman's "How to Prove It: A Structured Approach".
Proofbooks are memebooks, your time is better spent on actual maths books.

[Anonymous No.10991169]

>>10990906
>For any i, j
j = n+1

[Anonymous No.10991222]

>>10991024
Velleman is a fucking meme. Anyone who needed to use a book to learn how to do proofs will never have a future in mathematics.

[Anonymous No.10991227]

>>10991169
hahaha so EPIC!XD. an o n used le implicit index set :))) i made funny joke!!!!:) xD
go fuck yourself you human piece of trash. it's fucking obvious what i meant. lord almighty.

[Anonymous No.10991229]

>>10991227
>go fuck yourself you human piece of trash.
Do you really need to swear?

[Anonymous No.10991237]

>>10991227
>why yes i do get triggered by posts online how did you guess

[Anonymous No.10991787]

Should I learn all the theorem proofs I come across in a text?

[Anonymous No.10991806]

>>10991787
Depending on the theorem, you should have an outline in your head of how it's proved or just know the relevant facts needed to prove it.

[Anonymous No.10992320]

number theory best theory.

[Anonymous No.10992354]

>>10991787
Depends on the text and the theorems and your reasons for reading it.

[Anonymous No.10992505]

This problem has me stuck (it's not from a book or anything I'm trying to prove it just because I thought of it, it may be a known and trivial result that I don't know because I'm a brainlet)
What is the LCM of a list of positive integers of length k that sum up to some L?
So [math] n_{1} + n_{2} + ... + n_{k} = L [/math], what is the LCM?
As K goes to L then the values of [math] n_{k} [/math] all get smaller and at k = L the LCM is 1 as all the values are 1. But then let K = L-1, and all hell breaks loose, let alone for some k = L - b, [math] b \in \mathbb{N} [/math].
How would I begin to work with this problem?

[Anonymous No.10992576]

>>10992505
Nevermind I think I solved it

[Anonymous No.10992578]

>>10992576
Post it here then

[Anonymous No.10992978]

>>10991787
Much more relevant then the details of the proof is it's central idea, but knowing all the theorems really is a good idea if you want to continue your studies in that area.

[Anonymous No.10993016]

>>10987515
Would anyone be able to explain to me what the asterisk (*) in the line that says g(v) = g(q) - g(p) means?

[Anonymous No.10993057]

>>10993016
It means what it says, it's just notation to define a modified map.

[Anonymous No.10993066]

>>10993057
You're saying that g* is different than g? Can you expound on that at all?

[Anonymous No.10993078]

>>10993066
>You're saying that g* is different than g?
[math]g_\ast[/math] is different from [math]g[/math] but is produced by modifying [math]g[/math].
[math]g_\ast[/math]'s name is based on [math]g[/math], instead of moving on to [math]h[/math] or whatever, to make it obvious at a glance that [math]g_\ast[/math] is based on [math]g[/math].
>Can you expound on that at all?
You have to be 18 to post here.

[Anonymous No.10993081]

>>10993078
Alright I think that makes sense, thanks

[Anonymous No.10993185]

What do they mean when factoring the x^2? Are they saying that absolute value is irrelevant because positive or negative, x^2 will always be positive?

What about absolute value when factoring out an x^3? Do you then need to consider it?

[Anonymous No.10993194]

>>10993185

are you memeing here

seeing as you can;t be fucked to work it out by hand and you've apparently never taken a basic algebra class that dealt with quadratic and cubic equations

1. go to your calculator
2. Graph x^2
3. Graph x^3
4. tell us what you see

fucks sake. what are you doing in an intro calc class?

[Anonymous No.10993205]

>>10993194
because I took a placement test with ALEKS after being out of school for 10 years and did well and placed into calculus 1

[Anonymous No.10993210]

>>10993205
what and the test never asked you to factor out?
look, just look up the field axioms and you should trivially understand what's been done.

[Anonymous No.10993265]

>>10993185
Yes, the absolute value is not necessary because x^2 will always be positive.

[Anonymous No.10993451]

Hey /mg/ physicisticist here with linear algebra exam coming up...

Is there a straight forward way to prove linear independence like in these two exercises or do I just have to have dude intuition lmao? I know theres some sort of strategy where you add up all the given linear maps with real coefficients and prove that theres no way to get the zero element, but I have no clue how that would work for the second one. Is there even an actual straight forward way, like I remember from maths 1, where you had to know the way of proving continuity or convergence and the likes, but once you got the hang of epsilon proofs it was fairly staight forward and the only thing left was computation. If there is such a thing as well for linear algebra Id greatly appreciate if someone can point me into the right direction for a good resource on that.

I seem to be really bad at the whole looking at a problem and finding an intelligent solution, even though I can do that somewhat well when its about theoretical phyics, even though one of six questions will be just like that, as in a question that you can only answer intelligently cause youve never seen anything like it in class, which I think is a great way to prepare an exam, even though I definitely am not smart enough to profit from it...

[Anonymous No.10993468]

>>10993451
do you know the definition of linear independence? no offense but these are really simple

[Anonymous No.10993471]

>>10993451
>Is there a straight forward way to prove linear independence like in these two exercises
Assume they are linearly dependent -> algebraic manipulation contradiction.

Linear independence in the first case literally just means there is no c such that c times p = q.

For the second I don't even know what to tell you. If I were your TA I would accept "true by definition" as an answer.

You REALLY need to read through the definitions again.

>if someone can point me into the right direction for a good resource on that.
There is. It's called the definition of linear independence.

[Anonymous No.10993480]

>>10993451
This is my attempt at proving the second one with the "scheme", but I have no idea if thats rigorous enough...
The first one was part of the actual first exam, which I failed. Iirc I tried to prove it by claiming that every differentiable function can be constructed from an (infinite) sum of polynomials, so all the polynomials from zero to infinity form a linear base for the given space, which [math]t[/math] and [math]t^{2019}[/math] are obviously part of, like with a taylor expansion, and that trivially there is no linear operation that changes the degree of a polynomial or something like that, but that wasnt what the prof wanted and he gave me zero score (need like 18 in total to pass so thats pretty significant...)

>>10993468
Is it kinda what I did in pic related?
>>10993471
This really is my problem. I never got the grasp of learning definitions by memory and then actually applying them, but even now where I have the definitions right in front of me I struggle to see how to apply them. I know that this shit is supposedly easy as fuck and everyone but me seems to be able to pass it, whereas Im now gonna start my bachelor thesis while possibly failing completely cause Im stuck in second semester maths...

Also I dont think [math]c p = q[/math] is the solution, cause no way anyone can give six points for that and it doesnt use the fact that its differentiable functions and in the exam review we got to talk to the prof and he explained in great detail what I should have done, but that was weeks and a bunch of other exams ago so I dont remember everything.

[Anonymous No.10993499]

>>10993480
>Also I dont think c p = q is the solution
No, it is what you ASSUME is true and the bring to a contradiction.
Assume p and q are not linearly independent, then c * p = q then c * t = t^(2019) for all t then c = t^(2018) for all t, which is a contradiction.

For how you get c*p=q:
By definition of linear dependence (not linearly independence) there exists a_1 and a_2 such that a_1 * p + a_2 q = 0 so a_1/(-a_2) * p = q.
(The case if a_2=0, in which the division is not defined, is left as an exercise to the reader)

>Is it kinda what I did in pic related?
That is the correct solution, yes.

[Anonymous No.10993511]

>>10993499
Ok so I just downloaded S. Langs linear algebra, cause my script is pure suffering of notation I dont know, and with his definition got this solution for the problem.
So this really is the standard way of going about proving linear independence? And then its just a matter of reading the question and using the information to prove it.

Ill have to work through a bunch of the exercises in the book and hope that thats enough. Sadly have very little official material, cause this is literally the first semester where the exam actually looks like a maths exam, whereas over the last two decades it was just mindless computation of actual matrices with numbers in them and you had to find eigenvalues and shit like that where the only actual difficulty is not making computational errors...

[Anonymous No.10993523]

>>10993511
>So this really is the standard way of going about proving linear independence?
It is definitely a very common method.

And the solution works, although instead of forming derivatives, the exact same result can be gotten by simple algebraic computation without any calculus.

>Ill have to work through a bunch of the exercises in the book and hope that thats enough.
Do that, exercises are a good way to learn.

[Anonymous No.10993549]

What does evaluating a given matrix by "inspection" mean? I can't figure out what they mean with "by inspection" means :/

[Anonymous No.10993564]

>>10993549
Inspection means fiddle with the numbers and do some ad-hoc calculations to get to the desired result

[Anonymous No.10993604]

Is there a way to quickly determine subgroups of a Galois group and associated fixed fields?
I tried doing it for the polynomial [math]x^{5}-2[/math] and it literally took me 40 minutes to do the thing, and I made a couple of smaller mistakes along the way. It's relatively easy for me to determine all the automorphisms, I just look at the generators of the splitting field and write down all the combinations which pair up conjugates, but given that the group of the aforementioned polynomial is of order 20, determining subgroups and fixed fields is rather tedious.
Any tips?

[Anonymous No.10993607]

>>10993451
Retarded questions go in /sqt/.
And don't call yourself a physicist.

[Anonymous No.10993614]

>>10993564
thanks :)

[Anonymous No.10993678]

>>10993549
Generally solution/proof by inspection just means its pretty obvious when you look at it for a bit.

[Anonymous No.10993790]

>>10993604
I don't think there is an effective and general method.
I would generally try the following
1) find the order of group
2) find the subgroups of S_n which have this order
3) if there's more than one, do something to tell which one is the actual Galois group
For example, there is only one (up to isomorphism) subgroup of S_5 of order 20.

[Anonymous No.10993806]

>>10993480
>every differentiable function can be constructed from an infinite sum of polynomials
false. every analytic function can be.
>the polynomials form a linear base
false, they form a dense subspace of many spaces though. the linear span of the polynomials in C([0,1]) with the supnorm is dense. The polynomials (sort of) form a schauder basis for the space of analytic functions on a given open set. however, none of these are a basis - in a basis, you are only allowed to take finite sums to get to everything else. taking infinite sums requires a topology, which regular vector spaces do not have.
>there is no linear operator which changes the degree of a polynomial
lmao, false
the operator T(t^k) = t^{k+1} is a perfectly good linear operator, as is T(t^k) = t^{k-1} with T(1) = 0.
>that wasn't what the prof wanted
given that it made 3 incorrect assumptions, i'm leaning toward "it was wrong" over "the prof didn't get it."

[Anonymous No.10993970]

>>10993806
>false. every analytic function can be.
Nah. There is stone-Weierstrass.

>The polynomials (sort of) form a schauder basis for the space of analytic functions on a given open set.
Don't they even form a (not just sort of) Schauder Basis of L^p under certain conditions?

[profinite No.10993980]

I'm very bad at wrapping my head around topological definitions and their implications - why are profinite groups important, how should I think about them?

[Anonymous No.10994068]

This is a proof that any generalised eigenspace has a basis of Jordan chains.

I don't understand the induction. Specifically, why does the induction hypothesis hold for W?

[Anonymous No.10994143]

Any thoughts on this book ? Also is introductory point-set topology doable at precalculus level with some complementary research ?

[Anonymous No.10994171]

>>10994143
>Also is introductory point-set topology doable at precalculus level with some complementary research ?
Yes, but you shouldn't.

[Anonymous No.10994214]

>>10994143
>Also is introductory point-set topology doable at precalculus level with some complementary research ?
Yeah, but why would you do this? First become familiar with proofs and rigor and then do topology. You should do some calculus before at least.

[Anonymous No.10994231]

>>10994068
>Specifically, why does the induction hypothesis hold for W?
Think about it: By definition, any element in W is of the form [math]w = (\phi-\lambda id)v[/math] for some [math]v \in \tilde E_{\phi}(\lambda) = \ker((\phi-\lambda id)^k)[/math].
Therefore, [math](\phi-\lambda id)^{k-1} w = (\phi-\lambda id)^k v = 0[/math] for any [math]w \in W[/math]. Hence the induction hypothesis applies to W

[Anonymous No.10994258]

>>10994143
>Also is introductory point-set topology doable at precalculus level with some complementary research ?
Why don't you try it and find out?

[Anonymous No.10994267]

>>10994258
based annoying phrase poster

[Anonymous No.10994279]

Does anyone recommend Feller's probability volumes 1&2 ?

[Anonymous No.10994319]

>>10994171
>>10994214
>Yeah, but why would you do this? First become familiar with proofs and rigor and then do topology. You should do some calculus before at least.

I want to get some exposure to the subject, and I think it might be interesting to do both at the same time, it gives you some motivation to study a subject you're interested in and then do the research necessary to understand it.

>>10994258
I think I'll do that I have the book already, just wanted some pointers.

[Anonymous No.10994350]

>>10994319
>general topology
>interesting
>gives motivation

[Anonymous No.10994373]

>>10994319
>it gives you some motivation to study a subject you're interested in and then do the research necessary to understand it.
wtf are you talking about dude? Just do calculus first like a normal person.

[Anonymous No.10994374]

>>10994350
What does he mean by this ?

[Anonymous No.10994379]

>>10994374
>What does he mean by this ?
I'm not a "he".

[Anonymous No.10994387]

>>10994379
Yikes.

[Anonymous No.10994394]

>>10994379
Sure

[Anonymous No.10994395]

>>10994374
He means that general topology is too abstract for it to motivate anything. I'm pretty sure most people do real analysis before topology because it helps motivate topology.

[Anonymous No.10994397]

>>10994374
He's wondering how did you come up with the idea that general topology is interesting in and of itself, or that it can provide motivation for its own study.
General topology is a formalism that you have to grind through to study actually interesting and motivated subjects, such as functional analysis, differential topology, algebraic topology, piecewise linear topology, pointless topology, measure theory, k-theory, geometric topology, etc.

[Anonymous No.10994475]

>book is called "topology without tears"
>read it
>cry

why are mathematicians like this

[Anonymous No.10994482]

>>10994475
That's a total brainlet book tho
The comments of indians saying they used it on their masters are just sad.
You lack the mathematical maturity to tackle topology, it's fine.
Go work on something easier (but that pushes you) for a bit

[Anonymous No.10994512]

>>10993607
Cmon theres worse questions in this thread. Also I called myself a physicisticist...

>>10993806
>>10993970
>stone-Weierstrass
Ah yes the prof mentioned that when we went over the exam (honestly was very surprised that he took the time to go over it with every single student...).
In a way the corrector "didnt get it" really just because he most likely saw that it wasnt what was written on the answer sheet and just disregarded it. Prof said something along the lines of if I had actually mentioned stone-weierstrass and provided a bit more proof he might have given me a better score but that it was just too little.
Anyways I kinda understand what to do now so I hope I dont have to resort to improvised fuckery for the next try...

[Anonymous No.10994574]

>>10994379
proof? this is a science and math board after all.

[Anonymous No.10994718]

So if a function is continuous through a, then lim x-->a f(x) = f(a). How miniscule can the continuity be? If you have x defined from 0.999999999 through 1.0000000001, you could use direct substitution and say they are asking for x--->1, is there a rule about how small the continuity can be?

[Anonymous No.10994755]

>>10994718
If you're thinking of continuous functions on am interval, then it doesn't matter.
But you can have functions that ate continuous at only a single point.
Consider f(x) = x if x is rational, f(x) = 0 if x is irrational. Then f(x) is continuous at zero.

[Anonymous No.10994768]

If I have a system of linear equations of the form [math]d_i=a_k+a_p[/math] where [math]d_i[/math] are given integers. Is it true that if al the [math]d_i[/math] are even, then there must exist an integer solution?

[Anonymous No.10994779]

>>10994768
> integer solution
To what? What's the variable you want an integer solution to? What are a_k and a_p?

[Anonymous No.10994794]

>>10994779
The a_k and a_p are free variables. For example, something like 1=x+y, 2=x+z, 3=z+y

[Anonymous No.10994849]

>>10994794
So [math]a_i=x_i+x_{i+1}[/math], where [math]i \in Z_n[/math].
That seems like a shitty problem, I'll leave it to another anon.

[Anonymous No.10994859]

[eqn]\lim_{n\rightarrow \infty}(\frac{n + 1}{n}) = 2 \iff \left | \frac{n + 1}{n} -2\right | < \varepsilon, \forall \varepsilon > 0, \exists n \geq N \in \mathbb{N}.[/eqn]
[eqn]\left | \frac{n + 1}{n} -2\right | = \frac{n + 1}{n} - 2 = \frac{n}{n} + \frac{1}{n} - 2 = \frac{1}{n} - 1 < \varepsilon \iff \frac{1}{n} < \varepsilon + 1 \iff n > \frac{1}{\varepsilon + 1}[/eqn]
Therefore, [math]N > \frac{1}{\varepsilon + 1}[/math].

So, assume[math]\varepsilon > 0, n \geq N \exists N \in \mathbb{N} | N > \frac{1}{\varepsilon + 1}[/math].
Then, [math]N > \frac{1}{\varepsilon + 1} \iff \varepsilon + 1 > \frac{1}{N} \iff \varepsilon > \frac{1}{N} - 1 \iff \frac{N + 1}{N} - 2 < \varepsilon \iff \left | \frac{N + 1}{N} - 2 \right | < \varepsilon[/math].
Therefore, [math]\lim_{n\rightarrow \infty}(\frac{n + 1}{n}) = 2. \square[/math]

Struggling to find the flaw in logic here.

[Anonymous No.10994864]

>>10994859
Second line, the first equality is wrong.

[Anonymous No.10994866]

>>10994859
Didn't mean to add the existence operator in the first line, that doesn't need to be there I don't think

[Anonymous No.10994868]

>>10994864
How? n is a natural number so the the value will always be positive and so the negative part of the absolute value can be ignored

[Anonymous No.10994871]

>>10994868
Set n=3 and test it.

[Anonymous No.10994884]

>>10994871
Ok, ok, so is that what immediately breaks the proof? We can't assume the absolute value will always be less than epsilon, and therefore no relationship between N and epsilon can be determined?

[Anonymous No.10994899]

>>10994768
Any halfway decent number theory book has conditions for the existence of solutions for systems
[math]d_i = a_ix+b_iy | i\in \{1,2,...,n\}[/math]

[Anonymous No.10994905]

>>10994884
>absolute value
>being smaller than 0

[Anonymous No.10994906]

>>10987515

>urge to explicitly calculate using elementary arithmetic, pen and paper intensifies

[Anonymous No.10994916]

>>10994884
>that what breaks the proof?
Lad, have you ever heard of ekusplojon? There is no such thing as "which mistake breaks the proof".

[Anonymous No.10994923]

>>10994899
Would you at least name a book? I'm not taking a course, the problem just came up in other shit I was doing.

[Anonymous No.10994947]

>>10994923
I like Leveque's Fundamentals of Number Theory.
Stein's Elementary Number Theory has it too.

[Anonymous No.10994975]

>>10994905
>>10994916
Just tell me how to do my goddamn homework you labrats

[Anonymous No.10994977]

Does anyone have access to a PDF of H. Toda's Composition Methods in Homotopy Groups of Spheres? It's surprisingly hard to find online without institutional access, had no luck with libgen/scihub

[Anonymous No.10995045]

Consider a function that jumps at x=100 by 5:

f(x) := x for x<=100, else x+5

On the standard topology of R, how does the abstract continuous definition (inverse images of open sets are open) show that this is discontinuous?
All examples I can think of that should show that f is discontinuous fail.

[Anonymous No.10995068]

>>10995045
>All examples I can think of that should show that f is discontinuous fail.
Show your work.

[Anonymous No.10995070]

>>10995045
[math]f^{-1}[(99, 101)]=(99, 100][/math]

[Anonymous No.10995078]

>>10995068
The only interesting open interval to look at it (105, 110), say, and it's preimage is open too.

[Anonymous No.10995085]

>>10995070
why is it defined it I take an interval containing 100.7?

[Anonymous No.10995087]

>>10995076
>(99, 100] union (100, 101)
>not an open or closed set

[Anonymous No.10995090]

>>10995085
What?
The preimage isn't restricted to the image of the function.

[Anonymous No.10995095]

>>10995090
k, that was my error.
I see the light now

[Anonymous No.10995098]

>>10995045
Look at [math](105-\varepsilon, 105+\varepsilon)[/math]. It's preimage will be [math](105-\varepsilon, 105+\varepsilon)\cup\{100\}[/math], and this should make you sad.

[Anonymous No.10995099]

>>10995095
>I see the light now
Mathematicians use "we", not "I".

[Anonymous No.10995104]

>>10995099
I certainly never use "I"

[Anonymous No.10995105]

>>10994947
I don't find any theorem that discusses that. It's a problem of m linear Diophantine equations but in n variables. I understand that I can compute the smith normal form, but for the particular equation I have, the books kinda just says that it has a solution without much justification (And it mentions nothing from number theory). The extra restrictions are that each equation only contains 2 variables whose coefficients are either 1 or -1. So the question is if such a system of equations is solvable when the "d_i"s are all even.

[Anonymous No.10995114]

>>10995087
Brainfart

[Anonymous No.10995136]

>>10995098
How is 100 in it, which is mapped to by 100.

Also, what is the cardinality of the standard topology on R?

[Anonymous No.10995158]

>>10995136
R is second countable, i.e. it has a countable base T.
Any open set can be written as a union of sets in T, so the topology accepts a surjection from P(T) and has cardinality lower than aleph one.
If we take (n, n+1)->n, and we consider arbitrary unions of the (n, n+1) under the exact same logic from earlier, we have a backwards surjection.
So it should be aleph one, but I suck at sets.
>>10995114
Relatable.

[Anonymous No.10995179]

>>10995136
Yeah, I misread the function.
> Consider a function that jumps at x=100 by 5, ...
Could mean a few different things if you don't actually read the function.

[Anonymous No.10995193]

>>10987515
There is easier way around, there is possible to make generator that spits out those numbers.

[Anonymous No.10995244]

>>10990473
I think they mean that there's a 31% chance of precipitation happening that day.

[Anonymous No.10995278]

>>10987515
hey guys, I just solved the sum-of-three-cubes for the number -3

[math]-569936821221962380720^3+(569936821113563493509)^3+(472715493453327032)^3 = -3[/math]

[Anonymous No.10995490]

>>10993970
Stone Weierstrass is for compact spaces, right? Does it hold on R? I doubt it.
In any case, any convergent power series is analytic on its domain of convergence by definition. I don't really see how you plan on getting anywhere outside of that.
>Under certain conditions
Of course, with the right norm. But who gives a shit about that. Christ.
>>10994512
Your professor was being nice. If you had put Stone Weierstrass on the exam, you'd still be wrong, because again, it implies absolutely nothing about the linear independence of t and t^2019. You need to KNOW that they are linearly independent to PROVE they're part of a Schauder basis to begin with. You can't use something that uses what you are trying to show.
>>10994718
There is no rule about how small the continuity can be, except that it has to be "some size." i.e, for every epsilon > 0, there EXISTS a delta > 0. But delta can be as small as you like.
>>10994859
>>10994975
what in the fuck do you think the limit should be? jesus christ. IF I GIVE YOU 1000001/1000000 THEN IS THAT CLOSE TO FUCKING 2?
do the same proof but with a 1 there instead of a 2 and it works.

[Anonymous No.10995494]

>>10995158
>so it should be aleph one
lol

[Anonymous No.10995500]

Why isn't the set notation used in statistics/probability theory? Say you have a random variable X. which is basically a set. So why can't you writhe X={1,2,3...}

[Anonymous No.10995502]

Any good resources for elementary analysis? My teacher is really enthusiastic but also extremely ESL and I'm having some trouble following the textbook

[Anonymous No.10995533]

>>10988675
>Corollarilally
No one in the history of the internet except you has made this mistake

[Anonymous No.10995554]

I need some help bros.

My analysis course uses a fairly useless textbook and my prof goes over material at a blinding pace.

She assigns like 12-15 problems a week. I can understand the content in the textbook but when I try to approach the problems, I don't even know where to begin.

It also doesn't help that the textbook doesn't provide many examples or worked out solutions to learn from.

What should I do?

[Anonymous No.10995566]

>>10995554
First off sit for 2 hours with one exercise without killing yourself
Then if you don't succeed with it, look at people working similar stuff out on youtube

[Anonymous No.10995593]

>>10995533
Oh it's not a mistake. Try saying it out loud.

[Anonymous No.10995623]

>>10995593
Well it's not a word
Although saying Corollarilalooley is fun to do

[Anonymous No.10995644]

"there is a natural number such that if it is divisible by 3 then the next natural is also divisible by 3"

how do you prove this is true or false? my textbook claims it's true but i have no idea how they got that conclusion
how do you prove or disprove this statement?

[Anonymous No.10995646]

>>10995644
The existence of any number divisible by 9 proves that that's true

[Anonymous No.10995650]

>>10995644
assume 3 divides 1. then since 1 divides 2, 3 also divides 2.

[Anonymous No.10995653]

>>10994977
What're you looking for from it?
If you haven't seen Strickland's matlab plugin I'd take a look at that.

[Anonymous No.10995666]

>>10988630
My start was terrible. I got depressed and failed out of engineering. Started again at community college a few years later and found a real love for mathematics there.

[Anonymous No.10995672]

>>10995646
can you explain a bit more. i'm just not getting it. maybe i just go to sleep and ask the professor tomorrow.

[Anonymous No.10995674]

>>10995644
>>10995672
The explanation is I misunderstood the question. Try something to do with coprimes and modulos and whatnot. I suck at this stuff

[Anonymous No.10995757]

>>10995644
True. The number 4 exhibits this. If 4 were divisible by 3, then 5 would also be divisible by 3. Thing is, 4 isn't divisible by 3 so we have nothing to worry about. Vacuously true.

[Anonymous No.10995777]

>>10990706
what the fuck is R(X)/(X2+1) if not purely algebraic?
but to answer >>10990525 i never heard of one either, but we proved it in the last 5 minutes of the first topology class

[Anonymous No.10995800]

infinitism is a joke

[Anonymous No.10995803]

>>10995490
>Does it hold on R?
Nah, of course not.

>any convergent power series is analytic
Yeah, but Stone-Weierstrass doesn't give a powerseries, it gives a completely different "sum of polynomials" which just isn't a power series.

[Anonymous No.10995806]

>>10994718
>is there a rule about how small the continuity can be?
Yes. It's the Lipschitz Constant of the function, which can be potentially arbitrarily large.
For a differentiable function it is the maximum of it's derivative.

[Anonymous No.10995813]

>>10995644
If 1=3k then 1+1=6k=2*3k

[Anonymous No.10995925]

>>10995777
He means that if you use R (or a ring with it), you'll tend to freely make R axioms and not treat it as some generic field of characteristic 0

[Anonymous No.10996039]

>>10995777
R is not an algebraic object in the first place

[Anonymous No.10996064]

>>10987808
If you need to use cubical measuring cups to measure out exactly 3 L of Mountain Dew. And you can't use them more than three times.

[Anonymous No.10996066]

are there any new and good ways to evaluate graph clusters? it seems to be too situational and all the common ones, purity, mutual information, silhouette, etc have cons so its hard to rely on them for different clusters

[Anonymous No.10996420]

>>10995803
>nah, of course not
oh, duh. all polynomials are unbounded. yikers.
>SW doesn't give a powerseries
yes, that's my point
>>10994718
>>10995806
Ah, yes. Of course, I forgot about the modulus of continuity. That's what you're probably looking for.

[Anonymous No.10996465]

yeah but what about more important famous problems like Riemann hypothesis?

[Anonymous No.10996694]

>>10996465
terence tao will prove the following theorem in 3 years: for almost all complex numbers they almost satisfy the riemann hypothesis

[Anonymous No.10996703]

>>10996694
>the set of parallel universes where the Riemann Hypothesis is false is meager

[Anonymous No.10996755]

How much and what would I need to learn in order to understand a proof of this?
https://en.wikipedia.org/wiki/Hasse%27s_theorem_on_elliptic_curves
(i have a good understanding of pure math undergrad curriculum but no specific knowledge towards algebraic geometry)

[Anonymous No.10996870]

>>10996755
>How much and what would I need to learn in order to understand a proof of this?
Literally just a bit of analysis and the elliptic curve theory leading up to it, the proof is really simple (and very pleasant).

[Anonymous No.10997049]

If [math]K/F[/math] is a finite Galois extension, then by the primitive element theorem there is an [math]\alpha \in K [/math] such that [math]K=F(\alpha)[/math]. My question is, do [math]n[/math] different zeroes of [math]irr(\alpha,F)[/math] generate [math]K[/math] as a vector space over [math]F[/math]? My guess is that they do, because I suspect that they are linearly independent over [math]F[/math]. But I still can't prove that.
Anons pls help me out here.

[Anonymous No.10997057]

Essentially, the situation is the following:
Analytic geometry(the babby kind, not the complex kind) was added to the list of subjects I have to take for my degree out of nowhere. My country's government has a law which essentially lets me force some three professors into forming a committee, throwing a test at me, and letting me dodge taking the subject if I score high enough.
But the entire process is an absolute pain for them, so they'll throw the single hardest analytic geometry test in existence at me.
Anyhow, I need recommendations in texts about analytic geometry. Also, do you think I should brush up on algebraic and incidence geometry? I'll already be reviewing linear algebra, calc, diffgeo and good ole euclidean.
Hopefully I can just solve it by talking with them, tho.

[Anonymous No.10997075]

>>10987515
what was so hard
just solve z = (-x^3 - y^3 + 3)^(1/3), x (-x^3 - y^3 + 3)^(2/3) - y (-x^3 - y^3 + 3)^(2/3) + y^2 (-x^3 - y^3 + 3)^(1/3) + x^2 y - x^2 (-x^3 - y^3 + 3)^(1/3) - x y^2 != 0

[Anonymous No.10997081]

>>10995554
Maybe pick up a decent textbook and work though it yourself?
You can always ask people online, there are discord servers where people readily answer your questions and offer help.

[Anonymous No.10997085]

>>10997049
X^2 + 1

[Anonymous No.10997088]

>>10997085
whoops
nice one
thanks for that

[Anonymous No.10997108]

>>10997075
>just solve z = (-x^3 - y^3 + 3)^(1/3), x (-x^3 - y^3 + 3)^(2/3) - y (-x^3 - y^3 + 3)^(2/3) + y^2 (-x^3 - y^3 + 3)^(1/3) + x^2 y - x^2 (-x^3 - y^3 + 3)^(1/3) - x y^2 != 0
Yeah. I mean why don’t you “just” do that?
Maybe because restating the problem doesn’t give an answer to the problems?

[Anonymous No.10997124]

>>10997108
I'm busy working on bigger, harder and more profitable problems to fund my NEETdom, only a brainlet would waste their genius on such a trivial problem

[Anonymous No.10997174]

>>10997124
>profitable problems
Big Yikes.

Hopefully you don’t “solve” them by restating them too...

[Anonymous No.10997218]

How the hell does one get through an extremely boring "maths" course ? I signed up for a heavily applied course that will not be named because I couldn't get into another course because of schedule complications and this crap is so boring. I sit down to do the HW and end up working on another class. The only thing I can think of is relatively high doses of adderall but I don't want to do that ...

[Anonymous No.10997222]

how do I stop being a retard

[Anonymous No.10997227]

>>10997174
P=NP
N=P/P
N=1
gibme my monie

[Anonymous No.10997231]

>>10997218
How applied are we talking about?
Place it in the following scale: linear programming -> numerical calculus -> finite element methods -> PDEs -> distributions -> integral equations.

[Anonymous No.10997236]

>>10997227
What if P=0?????
Noob

[Anonymous No.10997243]

>>10997218
Just put little effort in and work yourself to death a week before the exam.

There is no cure for boredom, except discipline.

[Anonymous No.10997246]

>>10987515
I barely pass Ordinary Diffeq with a B in a hard and condensed summer course, should I consider going for a math minor? And what material should I study if I want to take the Putnam exam?

[Anonymous No.10997255]

>>10997246
if you barely passed ODE how do you think you're going to be qualified for the Putnam?

[Anonymous No.10997259]

>>10997255
Well thats why Im asking. And the Putnam doesnt use math higher than Calculus excluding certain other branches

[Anonymous No.10997271]

>>10997231
It's basically a multivariable calc course, it's a physics course with a good deal of reading that I can barely get through.

>>10997243
Yeah I can definitely just cram for this plus the pressure will be on if I wait till then, which is now. I might wait longer... I was thinking about finding a physics kid who is taking a common pure math elective and setting up a homework deal ... in an ideal world.

[Anonymous No.10997291]

>>10997271
>multivariable calc
That's barely a 4 in the scale of appliedshittery, suck it up.

[Anonymous No.10997330]

>>10997255
>>10997291
Dont be a dick. /sci/ isnt elitist (unironically)

[Anonymous No.10997397]

When can you "solve for" in an equality with quotient groups? For example, if you have an endomorphism of a group [math]\phi :G\to G[/math], then we know that [math]G/ker(\phi) = Im(\phi)[/math] Is it then true that [math]G/Im(\phi)=ker(\phi)[/math]?

[Anonymous No.10997413]

Why is it not recommended to self learn and only go to college for exams?

[Anonymous No.10997424]

>>10997413
....that's basically college. No one learns in lecture since you have to do the problems yourself without much help in order to understand and master the material

[Anonymous No.10997427]

>>10997397
The expression [math]G/\mathrm{Im}(\phi)[/math] doesn't make sense since the image of phi does not live in G.

[Anonymous No.10997437]

>>10997427
I explicitly called it and endomorphism, and after that I still showed where the function defined. Not trying to be a dick bruh, but I think it's pretty clear the image is a subgroup of [math]G[/math] as well.

[Anonymous No.10997452]

>>10997413
It is highly recommended, but there are obvious benefits of going to class too. Sometimes, the professors have a unique way of developing what they are teaching, so getting some notes is sometimes necessary because definitions, theorems and whatever and what can you use to solve problems can change. There are some classes that really are amazing, and will keep you motivated. And while math is autism central, something plenty of people forget is that is important to have a good relation with your professors for them to spot you. They are not looking at the grades and sending emails to every A student.

[Anonymous No.10997455]

>>10997437
It seems to work for
phi_n(x) := x mod n
in Z

ker is all multiples of n, Z/ker is literally Z/nZ and the image is {0,1,2,...,n-1}.

Might be that this generalize to other endos that can be viewed as quotient constructions themselves - but I'm quickly running out of examples.

[Anonymous No.10997476]

>>10997437
Consider Z->nZ.

[Anonymous No.10997493]

>>10997476
>the absolute state of my brain
Anyhow, I think it actually worked since the inclusion of the image gives a retract and then split exact sequence applies. But that's for abelian groups, you're actually insane if you think it works for anabelian ones.

[Anonymous No.10997534]

I've been asked to prove that [math]t\geq 0 \mid t\in \mathbb{R} \to \exists n \in \mathbb{N} \mid n-1 \leq t < n[/math]. This is as far as I've gotten, but I feel like I'm missing a piece

[Anonymous No.10997541]

>>10997534
>he doesn't even mention the archimedean property in the proof
What are you even trying to do?

[Anonymous No.10997542]

>>10997541
The archimedian property is in this chapter, I'll double back and see if I can incorporate that into it

[Anonymous No.10997546]

>>10997397
No. Let G = (integers, +) and let phi: x -> 2x

[Anonymous No.10997554]

>>10997534
What does this statement even mean? This is what people mean when they say symbols shouldn't replace words.

[Anonymous No.10997559]

>>10997554
It means t is between two sequential natural numbers

[Anonymous No.10997577]

>>10997559
>For every positive real number t, there exists a natural number n such that [math]n - 1 \leq t < n[/math]
Is this what you're trying to say?

[Anonymous No.10997584]

>>10997577
Yes

[Anonymous No.10997789]

>>10996694
Ahhh... I see you felt the need to exalt yourself for having eaten of that tree in the middle of garden.

[Anonymous No.10997896]

https://arxiv.org/pdf/1909.10313
>A Proof of Riemann Hypothesis
>Tao Liu, Juhao Wu
>(Submitted on 19 Sep 2019)

>The ratio of the Riemann-Zeta function [math]W(s)=ζ(s)/ζ(1-s)[/math] maps the line of [math]s=1/2+it[/math] onto the unit circle in W-space. [math]|W(s)|=0[/math] gives the trivial zeroes of the Riemann-Zeta function [math]ζ(s)[/math]. In the range: [math]0&lt;|W(s)|\neq1[/math], [math]ζ(s)[/math] does not have nontrivial zeroes. [math]|W(s)|=1[/math] is the necessary condition of the non-trivial zeros of the Riemann-Zeta function. Writing [math]s=σ+it[/math], in the range: [math]0\leqσ\leq1[/math], but [math]σ\neq 1/2[/math], even if [math]|W(s)|=1[/math], the Riemann-Zeta function [math]ζ(s)[/math] is non-zero. Based on these arguments, the non-trivial zeros of the Riemann-Zeta function [math]ζ(s)[/math] can only be on the [math]s=1/2+it[/math] critical line; therefore the proof of the Riemann Hypothesis.

[Anonymous No.10997900]

Threadly reminder to work with physicists.

[Anonymous No.10997938]

>>10997896
Why they never consider that if it was this simple someone else would've done it by now?

[Anonymous No.10998012]

>>10997896
the typesetting is awful

[Anonymous No.10998039]

>>10998012
How'd he squeeze an html entity into his Tex, I'm impressed

[Anonymous No.10998053]

>>10997789
Hey Tooker, I'm glad you are back. What happened to your last thread? Did they ban you?

[Anonymous No.10998063]

>>10997896
>chinese names
I'd sooner expect a proof of the Riemann hypothesis from Peru or Uganda.

[Anonymous No.10998192]

>>10997222
trips of photo truths

[Anonymous No.10998216]

This is a very clean version, that fixed an error in the explanation from the January version, and made numerous readability and quality of life improvements. May make a full thread tomorrow.

[Anonymous No.10998224]

page 2

[Anonymous No.10998229]

p3

[Anonymous No.10998233]

Could do more, but thread is almost at 300. This version is down to 11 pages from 14.
If someone wants more, lemme know.

[Anonymous No.10998664]

Can someone tell me the natural isomorphism
between the associators functors?

[Anonymous No.10998800]

>>10998233
post all of the pages

[Anonymous No.10999062]

>>10998216
Are you claiming you solved the Riemann hypothesis?

[Anonymous No.10999070]

>>10996039
ok, you got me there

[Anonymous No.10999330]

[Anonymous No.10999336]

>>10999330
>We discuss the Riemann zeta function, the topology of its domain, and make an argument against the Riemann hypothesis.
>the topology of its domain
It's literally just the punctured plane, nothing interesting whatsoever. You can tell just by that one line that this thing is not worth even reading.

[Anonymous No.10999368]

>>10999330
Tooker, why the Gundam? I knew you were schizophrenic but are you also autistic? With how buff and rough you look, I never thought you'd be the kind to decorate his papers with shiny stickers. Do you have the 'tism?

[Anonymous No.10999554]

>>10998039
It's not even set in LaTeX, is it? It sure as hell doesn't look like it.
I thought it was just a word document at first but maybe they actually typeset the entire paper in fucking HTML.

[Anonymous No.10999871]

I just started my first year of uni, and the main courses are Linear Algebra, and Analysis.
These are the books I'm planning to read:
Lang, Linear Algebra
Dummit/Foote, Abstract algebra
Abbott, Understanding Analysis
Tao, Analysis
I'll probably try regardless, but do you guys think it's too much, or would you replace something in this list?

[Anonymous No.10999991]

>>10999871
If you have courses, why read books about the same thing? i'd rather do some exercises

[Anonymous No.11000017]

>>10999991
I will not be attending the lectures

[Anonymous No.11000019]

>>10999871
do more or you'll never make it

[Anonymous No.11000032]

>>11000017
why

[Anonymous No.11000036]

>>11000017
Retard

[Anonymous No.11000050]

>>11000032
It's a complete waste of time

[Anonymous No.11000055]

>>11000050
then why you enrolled in the first place

[Anonymous No.11000110]

>>11000055
Well having a degree is pretty important for finding work. Also internships. If I didn't go to college, I would probably have to work full time, which leaves little time for studying what I like. I think it's pointless to go to lectures when you have free access to texts written by experts in their field

[Anonymous No.11000114]

>>11000110
>degree for finding work
>math degree
Lad. You can`t just half sell out like that, either sell out the full way or don`t sell out at all.

[Anonymous No.11000149]

>>11000114
There's not much choice, as I doubt I'd be able to get into grad school if the profs only know me from exams. So full sell out is the plan, I'll go work at a trading firm. I'm not sure what you mean by half sell out

[Anonymous No.11000214]

>>11000050
You're in first year and you think you're gonna teach abstract algebra and real analysis to yourself. Go to the lectures, you dumb faggot

[Anonymous No.11000244]

>>11000214
>You're in first year and you think you're gonna teach abstract algebra and real analysis to yourself
The algebra part at least is quite easily done.

[Anonymous No.11000246]

Ok so I'm finally working on topological spaces. Can someone explain me why specific cases of topological spaces are interesting? Why do we need to study metric spaces or other more specific spaces when they are obviously specific examples of topological spaces? It feels really boring to study the specific definition of properties for each spaces when everything can be abstracted with a more general definition. Isn't math all about generalizing properties and structures? Why teach math from specific spaces to more general spaces instead of the other way around? Is it because it requires "mathematical maturity" to learn the most general definition of mathematical objects?

[Anonymous No.11000253]

>>10987515
>1+1+1
gee that sure was hard

[Anonymous No.11000255]

>>11000246
Because starting with the most abstract first is ass backwards and hinders intuition. Also maturity.

[Anonymous No.11000258]

>>11000214
But first year lectures wouldn't teach that material anyway

[Anonymous No.11000278]

>>11000258
>the absolute state of anon's university
>>11000246
Because general topological spaces have five or six properties.
>Isn't math all about generalizing properties and structures?
No. How does that sort of autism even happen?

[Anonymous No.11000287]

>>10997896
interesting. this is actually a meme i never understood with some of these people. what's the point of the proof if you aren't actually using any new tools or methods. then it's just boring

[Anonymous No.11000290]

>>11000278
>why isn't every university's first year coursework like math 55

[Anonymous No.11000321]

>>11000246
This isn't really a question, this is a shit take disguised as half a dozen questions, but I'll post responses to a few of them.
>Can someone explain me why specific cases of topological spaces are interesting? Why do we need to study metric spaces or other more specific spaces when they are obviously specific examples of topological spaces?
Because metric spaces are specific. You can say many things about metric spaces that you cannot say about general topological spaces. This is like asking why we bother studying calculus since continuous functions are just specific examples of arbitrary functions.

>Isn't math all about generalizing properties and structures? Why teach math from specific spaces to more general spaces instead of the other way around?
For one, because of the above; lots of common, fundamentally important spaces happen to be metric, and we need to prove results about them that aren't true or are substantially more difficult to prove in more general settings.
But the suggestion here is fundamentally wrong as well. A generalization only has any value in the first place because it generalizes something. Without knowing the examples that you're trying to abstract from, you're learning a generalization of nothing. It's meaningless.

Broadly speaking, the trade-off for increasing generality is that your theorems become less interesting. It's very hard to say anything interesting at all about completely general settings (I like a quote by Gromov here, which goes something like "anything statement proven for all finitely generated groups is either trivial or wrong")

[Anonymous No.11000329]

>>11000246
Well why not.
Is your complaint only that you consider it boring?

[Anonymous No.11000331]

I don't know why but I find large cardinals fascinating. Is this autism?

[Anonymous No.11000361]

>>11000321
Thank you for your answer, I finally realize that i'm autistic for aiming to ridiculously abstract useless things. I always found "simple" things more interesting even if not really usefull. Maybe I'm autistic enough for category theory or other trendy hipster mathematics (Hott).

[Anonymous No.11000369]

>>11000050
>>11000110
>>10999871
Just drop out. It’s inevitable anyway with that attitude.

>>11000258
My University taught some abstract Algebra as part of Lin Alg, some basic group theory and Galois theory.
In Analysis we did completion of metric spaces to construct the reals.

What you are doing is retarded. Go to the Lectures and IF you are noticing that you are blazing through them, THEN you can stop going and learn on your own.

[Anonymous No.11000389]

>>11000331
yes

[Anonymous No.11000399]

>>11000369
I went for a couple of days and was completely bored. But those were review, so it's expected. I guess I can go once every few weeks to see if I understand what they're doing. But I seriously doubt the coursework would outpace my self learning.

[Anonymous No.11000419]

>>11000399
Review is completely meaningless, a last effort to try to get the ones who are really far behind somewhat into the stuff.
For me it was a literal joke, while the lectures were somewhat of a challenge.

But you have to realise that advanced knowledge is built upon foundational knowledge, if you find the lectures boring try more to go more into the depth then ahead.
It is far more beneficial to know lin alg in depth, then a bit of real analysis.

[Anonymous No.11000425]

>>11000331
>set theory
yes, it's autism

[Anonymous No.11000456]

>>11000419
I would much rather study things that interest me. But unfortunately, those are require various prerequisites. So I only want to study the foundations enough to fullfil those requirements, as well as get a high grade.

[Anonymous No.11000469]

Can someone clarify domain and range for me? I understand the following:
>domain is x-value
>range is y-value
>domain can't give out two outputs
are there any other points or concepts im forgetting? I looked into the basic mathematics thing by lane, but there's nothing on this topic.

[Anonymous No.11000480]

>>11000456
>So I only want to study the foundations enough to fullfil those requirements, as well as get a high grade.
Why do you think it's a good idea to rush through math? You're not going to succeed with this attitude.

[Anonymous No.11000483]

>>11000469
Domain is what goes into the function. Range is what comes out. They're just simple definitions, there aren't really any deep points to think about here.

[Anonymous No.11000494]

>>11000480
Well what then would you consider a healthy progression? If not knowledge that is enough to get a high grade, and study more advanced topics that build on it.

[Anonymous No.11000513]

>>11000456
That is a very bad idea. The best way to understand higher math is having a very solid foundation in the foundations.
Also, if you do not enjoy, or at least appreciate, the prerequisites you won’t enjoy the rest.

There is no place for someone with a limited understanding of linear algebra doing real analysis. The intuition you build about R^n is simply invaluable.

>>11000494
There is nothing you gain from doing higher math, if you haven’t a good foundation.
The “prerequisite knowledge” is practically irrelevant, just knowing the 15 LA theorems used in real analysis will gain you nothing, the intuition you gain from serious study of LA will.

Also studying foundations “for the exams” is simply retarded.

There is a good reason universities do things in a certain order, if you do not feel challenged trying to circumvent the order will gain you basically nothing, you will have to relearn everything anyway as you had no chance to understand it the first time.

[Anonymous No.11000518]

>>11000483
What I mean is, for instance, how would you "find" the domain for let's say, f(x) = x + 5? etc.

[Anonymous No.11000520]

>>11000518
plot the graph
can you draw lines perpendicular to the x-axis for every point on the y-axis so that the line intersects the graph?

[Anonymous No.11000522]

>>11000520
What? That just makes it all the more confusing. Why would I need to graph it?

[Anonymous No.11000532]

>>11000518
The formula x + 5 makes sense for all x. So the domain is the real numbers, unless otherwise specified.

When you're asked questions about domain, it will usually be something like "what is the domain of 1/(x-2)"? And you're supposed to point out that 2 is not in the domain because the function's not defined there.

[Anonymous No.11000544]

>>11000513
I'd like to interject for a moment.
While I'll concede that intuition and foundations are both extremely important, you have seriously misguided recommendations for building those.
Getting stuck, moving forwards nonetheless and returning later is fundamental for progress, and insisting on understanding everything at first reading will only burn you out and make you lose interest. Furthermore, books on more advanced topic typically feature short recaps of the prerequisites needed, and this gives an excellent opportunity for recalling what was seen previously. Finally, a marginal understanding of later subjects is often a great help in understanding what came previously, such as the residue theorem and de Rham cohomology, the intermediate value theorem and basic general topology, or a host of results in the differential geometry of curves and surfaces and Riemannian geometry.

[Anonymous No.11000552]

>>11000532
>makes sense for all x
>domain is the real numbers
>because the function's not defined here
Can you please elaborate? I'm 100% new to this and using specific terminology doesn't help. I understood your last statement, because you can't have a 0 in the denominator because you can't divide by 0.

[Anonymous No.11000579]

>>11000552
The lowest-level way you can think about this (which you seem to want) is that the domain is all the numbers for which the function definition makes sense. Your example of f(x) = x + 5 makes sense no matter what number you choose to plug in for x, so the domain is the entire x axis; "all numbers" so to speak.
Some functions will not make sense for all numbers; something like f(x) = 1/(x-2) makes sense for every number except for two, so the range is all numbers except two. A different example like [math]f(x) = \sqrt x[/math] doesn't make sense for any negative number since negative numbers won't have square roots, so the domain is only the numbers which are greater than or equal to zero.

[Anonymous No.11000582]

>>11000579
>so the range is all numbers except two.
*domain is all numbers except two
woops

[Anonymous No.11000659]

>>11000579
So when they ask for the domain, they are asking for an expression? for example, in your f(x) = 1/(x-2) example, the answer is going to be "x>2"? and in that case, in "interval notation", it would be [x, 2)U(2, x]?? (just learned this notation thing like a couple of hours ago)

[Anonymous No.11000754]

>Mathlet who Hasn't done math in three years
>Have to pass high school stochastic next year

Should i an Hero?

[Anonymous No.11001182]

>>11000754
>high school stochastic
Damn, I didn't do stochastic PDEs in high school. Luuucky.

[Anonymous No.11001192]

>>10987808
https://www.youtube.com/watch?v=GXhzZAem7k0

I was just about to post something a bout that.

Okay so I'm a brainlet, but when they go searching for these solutions, do they take 2 cubes, subtract them. Then subtract the number they want to find for the remainder, and then compare it to all possible cubes?

Like
10^3 + 9^3 - 12^3 = 1

10^3 + 9^3 = 1729

1729 - 1 = 1728( = 12^3)

So like when they run their programs are they calculating the cube of a number then subtracting it? Or do they pre-calculate the cube results then compare them?

[Anonymous No.11001420]

>>11001192
Follow the link in the OP, then at the bottom of the page follow the "Cracking the problem with 33" link for a description of the algorithm used. I can't post the link here because 4chan thinks it's spam.

[Anonymous No.11001430]

>>11001420
I'm guessing they figured it out already.

That all you need to do is add or subtract the results of cubed numbers.

Then subtract the number you want to find from the result of that, and match it with a cubed number.

[Anonymous No.11001470]

For an undergrad looking into doing a masters (in Canada), how important is research?

[Anonymous No.11001685]

For the 3 cube stuff, assuming they are using pre-calculated tables

30 digits x 30 is...
265,252,859,812,191,000,000,000,000,000,000 permutations?
31556952000000000 ns a year
8,405,528,512,772,431.2 years to solve with 1ghz 1 instruction per ns 1 core CPU.
we're on at least 4ghz CPUs so that's ONLY 2.1 quadrillion years with 1 core


Is my math there correct?

Would still take us 42 billion years even using the fastest super computer available now.

[Anonymous No.11001829]

>>11000544
I don’t really disagree, but the context here is missing.
This isn’t about diving into a subject you don’t have learned the prerequisites, this is about skipping first year math, the point where students are introduced to basic proof techniques and are supposed to gain some fundamental intuition about mathematics.

There is a severe difference between “let’s look at some more advanced topic” (which my professors regularly did, like a short introduction to Lebesgue Integration after the chapter on Riemann Integration, or which is often done in Seminars) and skipping first year university classes to learn real analysis by yourself.

[Anonymous No.11002030]

Why are there so many computational problems in my analysis exam?

[Anonymous No.11002064]

>>10988331
is Herstein good? It's the recommended text for my algebra course

[Anonymous No.11002067]

>>11002030
because there are probably computer scientist taking the course, dumbing it down

[Anonymous No.11002076]

>>11000214
you'll never make it if you can't self learn without going to lectures